I would agree with the post above that the song seems mean-spirited. Luci Baines was born in 1947, so she was only 16 when her father became President. She was 19 when she got married, and 21 when her father lost to Richard Nixon. Seems a little young to be a political target, or to be "the brains behind our foreign policy."

"Luci Baines" was written by Marvin Hamlisch (music) and Howard Liebling (lyrics). I suppose Liebling's "greatest hit" was "Sunshine, Lollipops, and Rainbows," written with Hamlisch. Hamlisch also wrote "Nobody Does It Better," "The Way We Were," and "What I Did for Love."
"Luci Baines" was on The Mitchell Trio's That's the Way It's Gonna Be, which was released on the Mercury label in August 1965.

Performers were Mike Kobluk, Joe Frazier, John Denver - Chad Mitchell had left the Trio by that time.

LUCI BAINES
(Howard Liebling (words), Marvin Hamlisch (music)
[as recorded by The Mitchell Trio on That's the Way It's Gonna Be (1965]

[Spoken}
Across this mighty land of ours, each state has its birds and flowers;
St. Louis has its Cardinals, and Baltimore its Orioles,
And Washington DC, too, has a specie (sic) of its very own:
A high-breasted front-runner who makes every [unknown] say:
"I wish I might fly away [unintelligible]

(sung)
Luci Baines, she is no Jackie, but then who complains,
She may be tacky, but she is the brains behind our foreign policy;
Who else but Luci could it be?
Lately she has grown addicted to the royal "we,"
She tolerates no impropriety when Bobby Baker comes to tea.

[spoken] It's been a pleasure to serve you, y'hear?

(sung)
Guess who all was there on my date with Luci Baines...
12 of Lynda's roommates,
11 Life reporters
10 Texas Rangers
9 Supreme Court judges
8 Stingray salesmen
7 Secret Service
6 Geese a-laying
Lyndon B. himself....
4 Beagles
3 Congressmen
2 Baptizing clergy
Luci Baines and me

Luci Baines, just like her pop
Her popularity gains.
She loves to sing out all those old refrains....

Oh, those TV towers of Texas, belong to Mom and me,
You'll find no Texas real estate in the name of Lyndon B....

The eyes of Bishop Pike are upon you...

My pop can make U Thant say no.

Pop said, "Tut-tut, we'll have to cut, deep in the heart of taxes...

Luci Baines, let other girls indulge - my girl abstains,
But you should see her on the backs of trains,
Why, she stands ten feet tall;
Oh, Lady Bird is sweet, and Lynda Bird's a tweet;
But Luci Baines, you're the biggest bird of them all.

Oh, what a bird.....

• The "baptizing clergy" - Luci was baptized Episcopalian, but became a Roman Catholic as a teenager. She received "conditional" baptism from the Catholics, just in case the Episcopalian baptism was invalid for some reason. The Episcopalians objected. Maybe Bishop Pike was one of those objecting, but I thought Bishop Pike had other issues on his mind.
• TV Stations and real estate: there was something about Lady Bird and the daughters owning TV stations and other assets, so LBJ could appear to be "clean."
• Burmese diplomat U Thant was Secretary General of the United Nations.
• Stingray Salesman: I think I recall seeing a photo of Luci driving a yellow Corvette Stingray convertible.
• Bobby Baker was a political advisor to LBJ, investigated for questionable financial dealings.